| L(s) = 1 | + 3-s + 3·5-s + 2·7-s − 2·9-s + 11-s + 13-s + 3·15-s + 2·19-s + 2·21-s − 3·23-s + 4·25-s − 5·27-s − 6·29-s − 31-s + 33-s + 6·35-s − 7·37-s + 39-s + 6·41-s + 8·43-s − 6·45-s + 12·47-s − 3·49-s − 6·53-s + 3·55-s + 2·57-s + 9·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 0.755·7-s − 2/3·9-s + 0.301·11-s + 0.277·13-s + 0.774·15-s + 0.458·19-s + 0.436·21-s − 0.625·23-s + 4/5·25-s − 0.962·27-s − 1.11·29-s − 0.179·31-s + 0.174·33-s + 1.01·35-s − 1.15·37-s + 0.160·39-s + 0.937·41-s + 1.21·43-s − 0.894·45-s + 1.75·47-s − 3/7·49-s − 0.824·53-s + 0.404·55-s + 0.264·57-s + 1.17·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.237215507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.237215507\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.71475943951933209828142912233, −9.653721086439290779296281041364, −9.066976572520307572209940611994, −8.231516309422736162372858340254, −7.23160735035634637227469842062, −5.92486237537320359558163673087, −5.43119443988629711656854867906, −3.98381349206586185709851546262, −2.62025296044033892131651679684, −1.63826190764885304516596012446,
1.63826190764885304516596012446, 2.62025296044033892131651679684, 3.98381349206586185709851546262, 5.43119443988629711656854867906, 5.92486237537320359558163673087, 7.23160735035634637227469842062, 8.231516309422736162372858340254, 9.066976572520307572209940611994, 9.653721086439290779296281041364, 10.71475943951933209828142912233
